Decoding and Error Correction In 2-D Arrays

ABSTRACT

A system and process for determining the location of a captured image from a larger image is described. A non-repeating sequence may be folded into a non-repeating array in which the array is unique for every sub window of a given size. The image of the sub window may be captured and its location determined within the non-repeating array.

CROSS REFERENCE TO RELATED APPLICATIONS

This is a divisional of application Ser. No. 10/284,422, filed Oct. 31,2002.

TECHNICAL FIELD

The present invention relates to interacting with a medium using adigital pen. More particularly, the present invention relates todetermining the location of a digital pen during interaction with one ormore surfaces.

BACKGROUND

Computer users are accustomed to using a mouse and keyboard as a way ofinteracting with a personal computer. While personal computers provide anumber of advantages over written documents, most users continue toperform certain functions using printed paper. Some of these functionsinclude reading and annotating written documents. In the case ofannotations, the printed document assumes a greater significance becauseof the annotations placed on it by the user. One of the difficulties,however, with having a printed document with annotations is the laterneed to have the annotations entered back into the electronic form ofthe document. This requires the original user or another user to wadethrough the annotations and enter them into a personal computer. In somecases, a user will scan in the annotations and the original text,thereby creating a new document. These multiple steps make theinteraction between the printed document and the electronic version ofthe document difficult to handle on a repeated basis. Further,scanned-in images are frequently non-modifiable. There may be no way toseparate the annotations from the original text. This makes using theannotations difficult. Accordingly, an improved way of handlingannotations is needed.

One technique of capturing handwritten information is by using a penwhose location may be determined during writing. One pen that providesthis capability is the Anoto pen by Anoto Inc. This pen functions byusing a camera to capture an image of paper encoded with a predefinedpattern. An example of the image pattern is shown in FIG. 11. Thispattern is used by the Anoto pen (by Anoto Inc.) to determine a locationof a pen on a piece of paper. However, it is unclear how efficient thedetermination of the location is with the system used by the Anoto pen.To provide efficient determination of the location of the capturedimage, a system is needed that provides efficient decoding of thecaptured image.

SUMMARY

Aspects of the present invention provide solutions to at least one ofthe issues mentioned above, thereby enabling one to locate a position orpositions on a viewed image. The viewed image may be on a printed imageor may be a graphical representation on a displayed computer image.Aspects of the present invention include a decoding process that permitsefficient decoding of a captured image, providing for efficientdetermination of the location of the image. These and other aspects ofthe present invention will become known through the following drawingsand associated description.

BRIEF DESCRIPTION OF DRAWINGS

The foregoing summary of the invention, as well as the followingdetailed description of preferred embodiments, is better understood whenread in conjunction with the accompanying drawings, which are includedby way of example, and not by way of limitation with regard to theclaimed invention.

FIG. 1 shows a general description of a computer that may be used inconjunction with embodiments of the present invention.

FIGS. 2A and 2B show an image capture system and corresponding capturedimage in accordance with embodiments of the present invention.

FIGS. 3A through 3F show various sequences and folding techniques inaccordance with embodiments of the present invention.

FIGS. 4A through 4E show various encoding systems in accordance withembodiments of the present invention.

FIGS. 5A through 5D show four possible resultant corners associated withthe encoding system according to FIGS. 4A and 4B.

FIG. 6 shows rotation of a captured image portion in accordance withembodiments of the present invention.

FIG. 7 shows various angles of rotation used in conjunction with thecoding system of FIGS. 4A through 4E.

FIG. 8 shows a process for determining the location of a captured arrayin accordance with embodiments of the present invention.

FIG. 9 shows a method for determining the location of a captured imagein accordance with embodiments of the present invention.

FIG. 10 shows another method for determining the location of capturedimage in accordance with embodiments of the present invention.

FIG. 11 shows a conventional representation of encoding space in adocument.

DETAILED DESCRIPTION

Aspects of the present invention relate to determining the location of acaptured image in relation to a larger image. The location determinationmethod and system described herein may be used in combination with amulti-function pen.

The following is separated by subheadings for the benefit of the reader.The subheadings include: terms, general-purpose computer, imagecapturing pen, encoding of array, decoding, error correction, andlocation determination.

Terms

Pen—any writing implement that may or may not include the ability tostore ink. In some examples, a stylus with no ink capability may be usedas a pen in accordance with embodiments of the present invention.

Camera—an image capture system that may capture an image from paper orany other medium.

General Purpose Computer

FIG. 1 is a functional block diagram of an example of a conventionalgeneral-purpose digital computing environment that can be used toimplement various aspects of the present invention. In FIG. 1, acomputer 100 includes a processing unit 110, a system memory 120, and asystem bus 130 that couples various system components including thesystem memory to the processing unit 110. The system bus 130 may be anyof several types of bus structures including a memory bus or memorycontroller, a peripheral bus, and a local bus using any of a variety ofbus architectures. The system memory 120 includes read only memory (ROM)140 and random access memory (RAM) 150.

A basic input/output system 160 (BIOS), containing the basic routinesthat help to transfer information between elements within the computer100, such as during start-up, is stored in the ROM 140. The computer 100also includes a hard disk drive 170 for reading from and writing to ahard disk (not shown), a magnetic disk drive 180 for reading from orwriting to a removable magnetic disk 190, and an optical disk drive 191for reading from or writing to a removable optical disk 199 such as a CDROM or other optical media. The hard disk drive 170, magnetic disk drive180, and optical disk drive 191 are connected to the system bus 130 by ahard disk drive interface 192, a magnetic disk drive interface 193, andan optical disk drive interface 194, respectively. The drives and theirassociated computer-readable media provide nonvolatile storage ofcomputer readable instructions, data structures, program modules andother data for the personal computer 100. It will be appreciated bythose skilled in the art that other types of computer readable mediathat can store data that is accessible by a computer, such as magneticcassettes, flash memory cards, digital video disks, Bernoullicartridges, random access memories (RAMs), read only memories (ROMs),and the like, may also be used in the example operating environment.

A number of program modules can be stored on the hard disk drive 170,magnetic disk 190, optical disk 199, ROM 140 or RAM 150, including anoperating system 195, one or more application programs 196, otherprogram modules 197, and program data 198. A user can enter commands andinformation into the computer 100 through input devices such as akeyboard 101 and pointing device 102. Other input devices (not shown)may include a microphone, joystick, game pad, satellite dish, scanner orthe like. These and other input devices are often connected to theprocessing unit 110 through a serial port interface 106 that is coupledto the system bus, but may be connected by other interfaces, such as aparallel port, game port or a universal serial bus (USB). Further still,these devices may be coupled directly to the system bus 130 via anappropriate interface (not shown). A monitor 107 or other type ofdisplay device is also connected to the system bus 130 via an interface,such as a video adapter 108. In addition to the monitor, personalcomputers typically include other peripheral output devices (not shown),such as speakers and printers. In a preferred embodiment, a pendigitizer 165 and accompanying pen or stylus 166 are provided in orderto digitally capture freehand input. Although a direct connectionbetween the pen digitizer 165 and the serial port is shown, in practice,the pen digitizer 165 may be coupled to the processing unit 110directly, via a parallel port or other interface and the system bus 130as known in the art. Furthermore, although the digitizer 165 is shownapart from the monitor 107, it is preferred that the usable input areaof the digitizer 165 be co-extensive with the display area of themonitor 107. Further still, the digitizer 165 may be integrated in themonitor 107, or may exist as a separate device overlaying or otherwiseappended to the monitor 107.

The computer 100 can operate in a networked environment using logicalconnections to one or more remote computers, such as a remote computer109. The remote computer 109 can be a server, a router, a network PC, apeer device or other common network node, and typically includes many orall of the elements described above relative to the computer 100,although only a memory storage device 111 has been illustrated inFIG. 1. The logical connections depicted in FIG. 1 include a local areanetwork (LAN) 112 and a wide area network (WAN) 113. Such networkingenvironments are commonplace in offices, enterprise-wide computernetworks, intranets and the Internet.

When used in a LAN networking environment, the computer 100 is connectedto the local network 112 through a network interface or adapter 114.When used in a WAN networking environment, the personal computer 100typically includes a modem 115 or other means for establishing acommunications over the wide area network 113, such as the Internet. Themodem 115, which may be internal or external, is connected to the systembus 130 via the serial port interface 106. In a networked environment,program modules depicted relative to the personal computer 100, orportions thereof, may be stored in the remote memory storage device.

It will be appreciated that the network connections shown areillustrative and other techniques for establishing a communications linkbetween the computers can be used. The existence of any of variouswell-known protocols such as TCP/IP, Ethernet, FTP, HTTP, Bluetooth,IEEE 802.11x and the like is presumed, and the system can be operated ina client-server configuration to permit a user to retrieve web pagesfrom a web-based server. Any of various conventional web browsers can beused to display and manipulate data on web pages.

Image Capturing Pen

Aspects of the present invention include placing an encoded data streamin a displayed form. The displayed form may be printed paper (or otherphysical medium) or may be a display projecting the encoded data streamin conjunction with another image or set of images. For example, theencoded data stream may be represented as a physical graphical image onthe paper or a graphical image overlying the displayed image or may be aphysical (non-modifiable) graphical image on a display screen (so anyimage portion captured by a pen is locatable on the display screen).

This determination of the location of a captured image may be used todetermine the location of a user's interaction with the paper, medium,or display screen. In some aspects of the present invention, the pen maybe an ink pen writing on paper. In other aspects, the pen may be astylus with the user writing on the surface of a computer display. Anyinteraction may be provided back to the system with knowledge of theencoded image on the document or supporting the document displayed onthe computer screen. By repeatedly capturing the location of the camera,the system can track movement of the stylus being controlled by theuser. The displayed or printed image may be a watermark associated withthe blank or content-rich paper or may be a watermark associated with adisplayed image or a fixed coding overlying a screen or built into ascreen.

FIGS. 2A and 2B show an illustrative example of pen 201 with a camera203. Pen 201 includes a tip 202 that may or may not include an inkreservoir. Camera 203 captures an image 204 from surface 207. Pen 201may further include additional sensors and/or processors as representedin broken box 206. These sensors and/or processors 206 may also includethe ability to transmit information to another pen 201 and/or a personalcomputer (for example, via Bluetooth or other wireless protocols).

FIG. 2B represents an image as viewed by camera 203. In one illustrativeexample, the field of view of camera 203 is 32×32 pixels (where N=32).Accordingly, FIG. 2B shows a field of view of 32 pixels long by 32pixels wide. The size of N is adjustable based on the degree of imageresolution desired. Also, while the field of view of the camera 203 isshown as a square for illustrative purposes here, the field of view mayinclude other shapes as is known in the art.

The input to the pen 201 from the camera 203 may be defined as asequence of image frames {Ii}, i=1, 2, . . . , A, where Ii is capturedby the pen 201 at sampling time ti. The sampling rate may be fixed ormay be variable based on the size of the document. The size of thecaptured image frame may be large or small, depending on the size of thedocument and the degree of exactness required. Also, the camera imagesize may be determined based on the size of the document to be searched.

The image captured by camera 203 may be used directly by the processingsystem or may undergo pre-filtering. This pre-filtering may occur in pen201 or may occur outside of pen 201 (for example, in a personalcomputer).

The image size of FIG. 2B is 32×32 pixels. If each encoding unit size is3×3 pixels, then the number of captured encoded units would beapproximately 100 units. If the encoding unit size is 5×5, then thenumber of captured encoded units is approximately 36.

FIG. 2A also shows the image plane 209 on which an image 210 of thepattern from location 204 is formed. Light received from the pattern onthe object plane 207 is focused by lens 208. Lens 208 may be a singlelens or a multi-part lens system, but is represented here as a singlelens for simplicity. Image capturing sensor 211 captures the image 210.

The image sensor 211 may be large enough to capture the image 210.Alternatively, the image sensor 211 may be large enough to capture animage of the pen tip 202 at location 212. For reference, the image atlocation 212 is referred to as the virtual pen tip. It is noted that thevirtual pen tip location with respect to image sensor 211 is fixedbecause of the constant relationship between the pen tip, the lens 208,and the image sensor 211. Because the transformation from the locationof the virtual pen tip 212 (represented by L_(virtual-pentip)) to thelocation of the real pen tip 202 (represented by L_(pentip)), one candetermine the location of the real pen tip in relation to a capturedimage 210.

The following transformation F_(S→P) transforms the image captured bycamera to the real image on the paper:L _(paper) =F _(S→P)(L _(Sensor))

During writing, the pen tip and the paper are on the same plane.Accordingly, the transformation from the virtual pen tip to the real pentip is also F_(S→P):L _(pentip) =F _(S→P)(L_(virtual-pentip))

The transformation F_(S→P) may be referred to as a perspectivetransformation. This simplifies as:$F_{S\rightarrow P}^{\prime} = \begin{Bmatrix}{{s_{x}\cos\quad\theta},} & {{s_{y}\cos\quad\theta},} & 0 \\{{{- s_{x}}\cos\quad\theta},} & {{s_{y}\cos\quad\theta},} & 0 \\{0,} & {0,} & 1\end{Bmatrix}$as the estimation of F_(S→P), in which θ, s_(x), and s_(y) are therotation and scale of two orientations of the pattern captured atlocation 204. Further, one can refine F′_(S→P) to F_(S→P) by matchingthe captured image with the corresponding background image on paper.“Refine” means to get a more precise perspective matrix F_(S→P) (8parameters) by a kind of optimization algorithm referred to as arecursive method. The recursive method treats the matrix F′_(S→P) as theinitial value. F_(S→P) describes the transformation between S and P moreprecisely than F′_(S→P).

Next, one can determine the location of virtual pen tip by calibration.

One places the pen tip 202 on a known location L_(pentip) on paper.Next, one tilts the pen, allowing the camera 203 to capture a series ofimages with different pen poses. For each image captured, one mayreceive the transform F_(S→P). From this transform, one can obtain thelocation of the virtual image of pen tip L_(virtual-pentip):L _(virtual-pentip) =F _(P→S)(L_(pentip))And,F _(P→S)=1/F _(S→P)

By averaging the L_(virtual-pentip) received from every image, anaccurate location of the virtual pen tip L_(virtual-pentip) may bedetermined.

The location of the virtual pen tip L_(virtual-pentip) is now known. Onecan also obtain the transformation F_(S→P) from image captured. Finally,one can use this information to determine the location of the real pentip L_(pentip):L _(pentip) =F _(S→P)(L_(virtual-pentip))Encoding of Array

A two-dimensional array may be constructed by folding a one-dimensionalsequence. Any portion of the two-dimensional array may be used todetermine its location in complete two-dimensional array. One concern isthat the image portion captured by a camera is limited in size.Accordingly, one needs to be able to determine the location from thecaptured image or a few captured images. Additional complications arisein that errors present in the captured image may compromise theavailability of some points in the captured image. So as to minimize thepossibility of a captured image portion being associated with two ormore locations in the two-dimensional array, a non-repeating sequencemay be used to create the array. One property of a created sequence isthat the sequence does not repeat over a length (or window) n. Thefollowing describes the creation of the one-dimensional sequence thenthe folding of the sequence into an array.

Sequence Construction

A sequence of numbers may be used as the starting point of the encodingsystem. For example, a sequence (also referred to as an m-sequence) maybe represented as a q-element set in field F_(q). Here, q=p^(n) wheren≧1 and where p is a prime number. The sequence or m-sequence may begenerated by a variety of different techniques including, but notlimited to, polynomial division. Using polynomial division, the sequencemay be defined as follows: $\begin{matrix}\frac{R_{l}(x)}{P_{n}(x)} & (1)\end{matrix}$where Pn(x) is a primitive polynomial of degree n in field F_(q)[x](having q elements). R_(l)(x) is a nonzero polynomial of degree l (wherel<n) in field F_(q)[x]. The sequence may be created used an iterativeprocedure using two steps: first, dividing the two polynomials(resulting in an element of field Fq) and, second, multiplying theremainder by x. The computation stops when the output begins to repeat.This process may be implemented using a linear feedback shift registeras set forth in an article by Douglas W. Clark and Lih-Jyh Weng,“Maximal and Near-Maximal Shift Register Sequences: Efficient EventCounters and Easy Discrete Logarithms,” IEEE Transactions on Computers43.5 (May 1994, pp 560-568). In this environment, a relationship isestablished between cyclical shifting of the sequence and polynomialR_(l)(x): changing R_(l)(x) only cyclically shifts the sequence andevery cyclical shifting corresponds to a polynomial R_(l)(x). One of theproperties of the resulting sequence is that, over a width (or length)n, any portion exists once in the sequence.

The process described above is but one of a variety of processes thatmay be used to create a non-repeating sequence.

Array Construction

The array (or m-array) that may be used to create the image (of which aportion may be captured by the camera) is an extension of theone-dimensional sequence or m-sequence. Let A be an array of period (m₁,m₂), namely A(k+m₁, l)=A(k, l+m₂)=A(k, l). When an n₁×n₂ window shiftsthrough a period of A, all the nonzero n₁×n₂ matrices over F_(q) appearonce and only once. This property may be referred to as a “windowproperty” in that each window is unique. A may then be expressed as anarray of period (m1, m2) (with m₁ and m₂ being the horizontal andvertical number of bits present in the array) and order (n1 , n2).

A binary array (or m-array) may be constructed by folding the sequence.One approach is to obtain a sequence then fold it to a size m₁×m₂ wherethe length of the array is L=m₁×m₂=2^(n)−1. Alternatively, one may startwith a predetermined size of the space one wants to cover (for example,one sheet of paper, 30 sheets of paper or the size of a computermonitor), determine the area (m₁×m₂), then use the size to let and L≧m₁m₂, where L=2^(n)−1.

A variety of different folding techniques may be used. For example,FIGS. 3A through 3C show three different sequences. Each of these may befolded into the array shown as FIG. 3D. The three different foldingmethods are shown as the overlay in FIG. 3D and as the raster pathsFIGS. 3E and 3F. The folding method as shown in FIG. 3D may be expressedas when gcd(m₁, m₂)=1 where L=2^(n)−1.

To create the folding method as shown in FIG. 3D, one creates a sequence{a_(i)} of length L and order n. Next, an array {b_(kl)} of size m₁×m₂is created from the sequence {a_(i)} by letting each bit of the array becalculated as shown by equation 1:b _(kl) =a _(i) where, k=i mod(m ₁), l=i mod(m ₂),i=0, . . . , L−1  (1)

This folding approach may be alternatively expressed as laying thesequence on the diagonal of the array, then continuing from the oppositeside when an edge is reached.

FIG. 4A shows sample encoding techniques for a first bit and a secondbit that may be used with the array of FIG. 3D. It is appreciated thatother encoding techniques may be used. For example, an alternativecoding technique is shown in FIG. 11.

Referring to FIG. 4A, a first bit 401 (for example, “1”) is representedby a column of dark ink. A second bit 402 (for example, “0”) isrepresented by a row of dark ink. It is appreciated that any color inkmay be used to represent the various bits. The only requirement in thecolor of the ink chosen is that it provides a significant contrast withthe background of the medium to be differentiable by an image capturesystem. The bits in FIG. 4A are represented by a 3×3 matrix of cells.The size of the matrix may be modified to be any size as based on thesize and resolution of an image capture system. Alternativerepresentation of bits 0 and 1 are shown in FIGS. 4C-4E. It isappreciated that the representation of a one or a zero for the sampleencodings of FIGS. 4A-4E may be switched without effect. FIG. 4C showsbit representations occupying two rows or columns in an interleavedarrangement. FIG. 4D shows an alternative arrangement of the pixels inrows and columns in a dashed form. Finally FIG. 4E show pixelrepresentations in columns and rows in an irregular spacing format(e.g., two dark blocks followed by a light block).

Referring back to FIG. 4A, if a bit is represented by a 3 by 3 matrixand an imaging system detects a dark row and two white rows in the 3×3region, then a zero is represented (or a one). If an image is detectedwith dark column and two white columns, then a one is represented (or azero).

Here, more than one block or pixel or dot is used to represent a bit.Using a single pixel (or block or bit) to represent a bit is fragile.Dust, creases in paper, non-planar surfaces, and the like createdifficulties in reading single bit representations of data units.However, it is appreciated that different approaches may be used tographically represent the array on a surface. Some approaches are shownin FIGS. 4C through 4E. It is appreciated that other approaches may beused as well. One approach is set forth in FIG. 10 using onlyspace-shifted dots.

A bit stream is used to create the graphical pattern 403 of FIG. 4B.Graphical pattern 403 includes 12 rows and 18 columns. The rows andcolumns are formed by a bit stream being converted into a graphicalrepresentation using bit representations 401 and 402. FIG. 4B may beviewed as having the following bit representation: $\begin{bmatrix}0 & 1 & 0 & 1 & 0 & 1 & 1 & 1 & 0 \\1 & 1 & 0 & 1 & 1 & 0 & 0 & 1 & 0 \\0 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 1 \\1 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 0\end{bmatrix}\quad$Decoding

When a person writes with the pen of FIG. 2A or moves the pen close tothe encoded pattern, the camera captures an image. The image is thenprocessed to determine the orientation of the captured image withrespect to the complete representation of the encoded image and extractthe bits that make up the encoded image.

FIG. 4B includes representations of the pixel blocks from image 403. Thedisplayed images 404-411 show 5×5 pixel blocks. Pixel block 404 shows adark row between white rows. Pixel block 405 shows a dark column betweenwhite columns. Pixel block 406 shows a bottom left corner. Pixel block407 shows a top right corner. The pixel block 408 shows a dark columnwith half a dark row on the left. Pixel block 409 shows a dark row withhalf a dark column above the row. The pixel block 410 shows half a darkrow. Pixel block 411 shows half a dark column. Analyzing the combinationof pixel blocks, one appreciates that all combinations of pixel may beformed by the image segments found in pixel blocks 404-411.

Further, one may attempt to represent all possible pixel combinationswith a smaller set of pixel blocks. Without more, one would expect thateach of the four blocks shown in FIGS. 5A-5D to be found in the image403. However, only three types of corners exist in the eight pixelblocks 404-411. In this example, there is no pixel block represented asFIG. 5A. By the fact that the image segments 401 and 402 may be chosento eliminate a type of corner, this presents the opportunity todetermine the orientation of a captured image based on the missing typeof corner.

Continuing to FIG. 6, the image a captured by a camera 601 may beanalyzed and its orientation determined so as to be interpretable as tothe position actually represented by the image 601. First, image 601 isreviewed to determine the angle θ needed to rotate the image so that thepixels are horizontally and vertically aligned. It is noted thatalternative grid alignments are possible including a rotation of theunderlying grid to a non-horizontal and vertical arrangement (forexample, 45 degrees). Using a non-horizontal and vertical arrangementmay provide the probable benefit of eliminating visual distractions fromthe user, as users may tend to notice horizontal and vertical patternsbefore others. For purposes of simplicity, the orientation of the grid(horizontal and vertical and any other rotation of the underlying grid)is referred to collectively as the predefined grid orientation.

Next, image 601 is analyzed to determine which corner is missing. Therotation amount o needed to rotate image 601 to an image ready fordecoding 603 is shown as o=(θ plus a rotation amount {defined by whichcorner missing}). The rotation amount is shown by the equation in FIG.7. Referring back to FIG. 6, angle θ is first determined by the layoutof the pixels to arrive at a horizontal and vertical (or otherpredefined grid orientation) arrangement of the pixels and the image isrotated as shown in 602. An analysis is then conducted to determine themissing corner and the image 602 rotated to present the image as 603 toset up the image for decoding. Here, the image is rotated 90 degreescounterclockwise so location of image 603 may be recovered.

It is appreciated that the rotation angle θ may be applied before orafter rotation of the image 601 to account for the missing corner.

Finally, the code in image 603 is read out and correlated with theoriginal bit stream used to create image 403. The correlation may beperformed in a number of ways. For example, it may be performed by arecursive approach in which a recovered bit stream is compared againstall other bit stream fragments within the original bit stream. Second, astatistical analysis may be performed between the recovered bit streamand the original bit stream, for example, by using a hamming distancebetween the two bit streams. It is appreciated that a variety ofapproaches may be used to determine the location of the recovered bitstream within the original bit stream.

Once one has the recovered bits, one needs to locate the captured imagewithin the original array (for example, the one shown in FIG. 4B). Theprocess of determining the location of a segment of bits within theentire array is complicated by a number of items. First, the actual bitsto be captured may be obscured from ideal capture by the camera (forexample, the camera may capture an image with handwriting that obscuresthe original code). Second, dust, creases, reflections, and the like mayalso create errors in the captured image. These errors make thelocalization process more difficult. In this regard, the image capturesystem may need to function with non-sequential bits extracted from theimage. The following represents a method for operating withnon-sequential bits from the image.

First, looking only at the sequence, let the sequence (or m-sequence) Icorrespond to the power series I(x)=1/P_(n)(x). Also, R(x)=r₀+r₁x +r₂x²+. . . +r_(n−1)x^(n−1) is any nonzero polynomial whose degree is lessthan n. Sequence R corresponds to R(x)/P_(n)(x). As mentioned above, Ris a shifted version of I. A distance D is defined as the distancebetween I and R. This may be represented as D(R)=s if R is the s-thshift of I. This distance D may be computed using a number of differenttechniques. One technique is by determining the Hamming distance betweenthe two. Another method is by using the Pohlig-Hellman-Silver algorithmas noted by Douglas W. Clark and Lih-Jyh Weng, “Maximal and Near-MaximalShift Register Sequences: Efficient Event Counters and Easy DiscreteLogorithms,” IEEE Transactions on Computers 43.5 (May 1994, pp 560-568).Other distance determining techniques as known in the art may also beused. Using the Pohlig-Hellman-Silver algorithm, R(x)≡x^(s)mod(P_(n)(x)).

However, attempting to compute a distance using the folded sequence ismore complex as the codes extracted may be non-consecutive and mayfurther include errors. Starting with the above sequence analysis, onemay determine a location from an image. One may start with the followingrelation:R=r ^(t) A  (2)where, r=(r₀ r₁ r₂ . . . r_(n−1))^(t), and A=(I T(I) . . .T^(n−1)(I))^(t) consists of the cyclic shifts of I. Accordingly, anycyclic shift of I is the product of r^(t) and A. If the pen reads k bits(b=(b₀ b₁ b₂ . . . b_(k−1))^(t) where (k≧n) of I), R(x) may be used tocyclically shift b₀ to the beginning of the sequence (or m-sequence).This shifting then adjusts the columns of A to create an n×k sub-matrixM of A. Hence,r ^(T) M=b ^(T).  (3)where T represents a translation of the matrix or vector.

If all the rows of M are independent, r may be determined and D(R)resulting in the absolute position of b₀ in I by modulating theone-dimensional locations of equation (1). This means that thetwo-dimensional location of the b₀ bit may be determined. This positioninformation permits the location of the pen tip to be determined.

Next, all modulations of A may be determined and checked.

Alternatively, because matrix A is large, not all modulations of A needto be retained. In fact, in some implementations, only relativepositions between the columns are relevant. Thus, the columns that maybe saved (after shifting b₀ to the beginning of the sequence (orm-sequence)) are those whose corresponding two-dimensional positions arewithin the reception field of the camera when the pen (and thus thecamera) rotates. The columns may be stored in an array withtwo-dimensional position information (e.g., a matrix). An exampleincludes:“element-type CoreVector [rowsize][column-size].”

The total number of such columns is much smaller than L (whereL=2^(n)−1). If b is error-free, the solution of r may be expressed as:r ^(t) ={tilde over (b)} ^(t) {tilde over (M)} ⁻¹  (4)where {tilde over (M)} is any non-degenerate n×n sub-matrix of M and{tilde over (b)} is the corresponding sub-vector of b.Error Correction

If errors exist in b, then the solution of r becomes more complex. Ifthe number of errors in b (expressed as n_(e)) is less than d/2, where dis the least Hamming distance of the code generated by M, b can becorrected. Unfortunately, as the matrix M changes for different penposes and that the structure of matrix M is not regular, traditionalmethods of decoding with error correction may not readily apply. In someinstances, it has been suggested that it is not possible to accuratelydetermine d when errors are present. See Ilya Dumer, Daniele Micciancioand Madhu Sudan, “Hardness of Approximating the Minimal Distance of aLinear Code”, IEEE Symposium on Foundations of Computer Science. 1999,pp. 475-485.

However, positive results have been achieved using a stochasticapproach. Assuming that n_(e) is relatively small compared to k, theprobability of choosing correct n bits from the k bits of b is largewhen randomly chosen n bits whose corresponding columns in M areindependent.

The system may repeatedly select random bits in an image. If the systemrepeats this selection and analysis a number of times, it can generallyisolate the proper distance in the sequence. In other words, whencorrect bits are chosen, the Hamming distance between b^(t) and r^(t)Mwill be minimal, where r is computed via equation (4). Repeated severaltimes, the correct r will occur more times than other distances and thuscan be easily identified. This result may be used alone or may becombined with previous location information. For example, if the cameracaptures a string of images that can be located in a line near the topof the page and the current image from the camera provides, for example,two distances from the previous string with one being very close and onebeing very far, it is more likely that the user only moved the pen asmall amount rather than a large amount in a very short time. Othertypes of analyses between previous image captures and current imagecaptures may be used.

If there are different r's such that the Hamming distance between b^(t)and r^(t)M are all minimal, then the probability that n_(e) exceeds theerror correcting ability of the code generated by M is large and thedecoding process fails. The system then may move onto the next capturedimage from the camera.

Finally, the correct b may be determined and the localization of r insequence (m-sequence) I can be computed by various techniques (includingbut not limited to the Pohlig-Hellman-Silver algorithm). The pen tip maythen be located.

FIG. 8 reflects a process that may be used to determine a location in asequence (or m-sequence) of a captured image. First, in step 801, a datastream relating to a captured image is received. In step 802, columnsare extracted from the captured image and a matrix constructed of theinformation in the captured image. In step 803, the code is sortedaccording to distance. The code may be sorted using a variety of sortingalgorithms as are known in the art. For example, a selection sortingalgorithm may be used. The selection sorting algorithm is beneficialwhere m is not large. However, if m becomes large, other sortingalgorithms (for example, a merge sort) that handle larger number ofitems more efficiently may be used.

In step 804, M row vectors are selected from the matrix constructed instep 802 and distance vector r is determined from the matrix of 802 tothe begging of the array (or m-array).

The captured image may include more information than needed to determinethe location of the matrix M. For example, if each bit is represented by9 pixels and the camera imaging system includes a 32×32 bit matrix, thenapproximately 100 bits may be obtained (10×10). Of these imaged bits, nare used to determine the location. So, for example, if the array m₁,m₂size was 255×255 (65,025), then the number of bits needed to locate acaptured image would be n where m₁×m₂≦L=2^(n)−1. Here, n=16.Accordingly, the number of bits the captured image would need to produceto be able to locate the image is n. In order to provide a variety ofdifferent selections to account for error, more than n bits need to beextracted from the image. For example, if the number of n needed is 16,the camera may be needed to produce 18 or more bits.

This analysis is performed K times (for example, 10 to 100 to 500 times)in step 805. The method attempts to determine the position that has thelargest number of repeated results and whose number of error bits isless than a value (for example, 3). The system then determines in step806 whether error correction was performed successfully. If no, then anerror is returned in step 809. If yes, then the position of theextracted code is calculated in the sequence (or m-sequence) in step 807(by, for example, the Pohig-Hellman-Silver algorithm). This step mayalso include the sub steps of calculating y_(i)=y_(n)/m₁, and looking upthe factors in the group table to get the residual value of R_(i). Thismay be expressed as: $\begin{matrix}{\left. {L = {\sum\left( {\frac{n}{mi} \cdot {Ri} \cdot {Vi}} \right)}} \right){mod}\quad n} & (5)\end{matrix}$

L represents the position in the sequence. Next, the (X, Y) position inthe array is expressed as x=L mod n₁ and y=L mod n₂ and the resultsreturned in step 808.

Location Determination

FIG. 9 shows a process for determining the location of a pen tip. Theinput is an image captured by a camera and the output may be a positioncoordinate of the image compared with the total image. Also, the outputmay include (or not) a rotation angle of the captured image.

In step 901, an image is received from a camera. Next, the receivedimage may be optionally preprocessed in step 902 (as shown by the brokenoutline of step 902) to adjust the contrast between the light and darkpixels, in an attempt to determine the rotation angle of the capturedimage, and the like.

Next, in step 903, the image is analyzed to determine the bit streamwithin it.

Next, in step 904, the bit stream is compared against the original bitstream and the location of the received bit stream within the originalbit stream is determined. This comparison may be made when the system isaware of how the original bit stream is wrapped to create the image 403.Finally, once the location of the captured image has been determined instep 904, the location of the pen tip may be determined in step 905.

FIG. 10 shows an alternative approach to determining the location of thepen tip. First, an image is received from the camera in step 1001. Theimage then may optionally undergo image preprocessing in step 1002 (asshown by the broken outline of step 1002). The pattern is extracted instep 1003. Here, pixels on the various lines may be extracted to findthe orientation of the pattern and determine the angle θ.

Next, the received image is analyzed in step 1004 to determine theunderlying distribution of pixels and grid lines. If grid lines arefound in step 1005, then the code is extracted from the pattern in step1006. The code is then decoded in step 1007 and the location of the pentip determined in step 1008. If no grid lines were found in step 1005,then an error is returned in step 1009.

Although the invention has been defined using the appended claims, theseclaims are illustrative in that the invention is intended to include theelements and steps described herein in any combination or subcombination. Accordingly, there are any number of alternativecombinations for defining the invention, which incorporate one or moreelements from the specification, including the description, claims, anddrawings, in various combinations or sub combinations. It will beapparent to those skilled in the relevant technology, in light of thepresent specification, that alternate combinations of aspects of theinvention, either alone or in combination with one or more elements orsteps defined herein, may be utilized as modifications or alterations ofthe invention or as part of the invention. It may be intended that thewritten description of the invention contained herein covers all suchmodifications and alterations.

1. A process for determining the location of a captured arraycomprising: extracting columns from said captured array; sortingextracted code from said columns by distance; constructing a matrix fromsaid columns and determining a vector from said code to code in a largerarray; and determining the position of extracted code.
 2. The processaccording to claim 1, further comprising: constructing at least anothermatrix from said columns and determining at least a second vector. 3.The process according to claim 1, further comprising: performing errorcorrection on said code.
 4. A computer-readable medium having a programstored thereon for determining the location of a captured array that,when executed by a computer causes the computer to: extract columns fromsaid captured array; sort extracted code from said columns by distance;construct a matrix from said columns and determine a vector from saidcode to code in a larger array; and determine the position of extractedcode.
 5. The computer-readable medium according to claim 4, furthercausing the computer to: construct at least another matrix from saidcolumns and determine at least a second vector.
 6. The computer-readablemedium according to claim 4, further causing the computer to: performerror correction on said code.
 7. A system for determining the locationof a captured array in a larger array comprising: a storage storinginformation regarding said larger array; an input receiving saidcaptured array; and a processor for determining a distance between saidcaptured array and said larger array by determining the shift of asequential version of said captured array from said larger array.
 8. Thesystem according to claim 7, wherein the vector providing the locationof said captured array in said larger array is given by:r ^(t) ={tilde over (b)} ^(t) {tilde over (M)} ⁻¹.